Learn Hardcore German: Im Hauptteil der Hausarbeit beschreiben wir unsere Methode genau. - In the main part of the term paper we describe our method precisely.

You can say In dem Hauptteil der Hausarbeit beschreiben wir …, and it is grammatically correct, but it sounds a bit heavier or more formal. In normal academic prose, im Hauptteil is the default.

What cases are Hauptteil and Hausarbeit in here, and why?

Break the phrase down:

im Hauptteil
- Hauptteil is masculine: der Hauptteil (nom. sg.)
- After in meaning a location (where?), German uses the dative.
- Masculine dative singular article is dem.
- in + dem Hauptteil → im Hauptteil
  → Hauptteil is dative singular masculine.
der Hausarbeit
- This is attached to Hauptteil and means of the term paper.
- In German, “the main part of the term paper” is expressed as der Hauptteil der Hausarbeit, with der Hausarbeit in the genitive, not with von.
- Hausarbeit is feminine: die Hausarbeit (nom. sg.).
- Feminine genitive singular article is der.
  → Hausarbeit is genitive singular feminine.

So the structure is:

im Hauptteil (dative)
der Hausarbeit (genitive, depending on Hauptteil)

Literally: In the main part of the term paper we describe our method in detail.

Why does the sentence start with Im Hauptteil der Hausarbeit instead of Wir?

German main clauses are verb-second (V2), not subject-first.

That means:

exactly one element comes first (the Vorfeld),
then the conjugated verb,
then the rest.

So both of these are correct:

Im Hauptteil der Hausarbeit beschreiben wir unsere Methode genau.
Wir beschreiben im Hauptteil der Hausarbeit unsere Methode genau.

The choice is about emphasis and information structure:

Starting with Wir → focuses on the actors (we).
Starting with Im Hauptteil der Hausarbeit → focuses on where/when in the paper this happens; very typical when describing the structure of a text.

In academic writing about the organization of a paper, it’s very natural to front the section phrase:

Im ersten Kapitel …
Im theoretischen Teil …
Im Hauptteil der Hausarbeit …

Can I also say Wir beschreiben im Hauptteil der Hausarbeit unsere Methode genau? Is there any difference?

Yes, that version is grammatically correct:

Wir beschreiben im Hauptteil der Hausarbeit unsere Methode genau.

Difference:

Im Hauptteil der Hausarbeit beschreiben wir …
→ Emphasis on the section of the paper. Good when the point is “In the main part, X happens.”
Wir beschreiben im Hauptteil der Hausarbeit …
→ More neutral topic order: subject first, then adverbial. Focus slightly more on what we do, less on where in the paper.

Content-wise they mean the same thing; the difference is mostly nuance in emphasis. Both are fine in academic German.

Where can genau go in the sentence, and do different positions change the meaning?

Common, natural options:

Im Hauptteil der Hausarbeit beschreiben wir unsere Methode genau.
Im Hauptteil der Hausarbeit beschreiben wir genau unsere Methode.

They are not fully identical:

… unsere Methode genau beschreiben
- genau modifies beschreiben.
- Meaning: “We describe our method in detail / precisely.”
- This is the usual interpretation here.
… genau unsere Methode beschreiben
- genau leans toward modifying unsere Methode.
- Nuance: “exactly our method (and not some other method)” or “this very method”.
- This version is possible, but in academic context it is less likely to be meant and can feel slightly marked in emphasis.

If you want to be very clear that the description itself is detailed, the safest is:

… beschreiben wir unsere Methode genau.
or even … beschreiben wir unsere Methode sehr genau / ausführlich / detailliert.

What is the function of genau here? Is it more like “exactly” or like “in detail”?

In this sentence, genau functions as an adverb of manner modifying beschreiben.

Two main uses of genau in German:

exact(ly) in the sense of precision of content:
- Welches Datum genau? – Which exact date?
- Genau diese Methode verwenden wir. – We use this very method.
closely / in detail / thoroughly when used with verbs like erklären, beschreiben, untersuchen:
- Wir erklären das Problem genau.
- Er hat den Ablauf genau beschrieben.

Here it is the second use:

… beschreiben wir unsere Methode genau.
→ We describe our method in detail / accurately / thoroughly.

Near-synonyms (more clearly “academic” in tone):

ausführlich (in a detailed way)
detailliert
präzise

So you could also write:

… beschreiben wir unsere Methode ausführlich.

Does Hausarbeit just mean “homework”, or something more specific?

Hausarbeit in a university context usually means:

a term paper / coursework essay / written assignment, often 10–20 pages or more,
written independently over several weeks,
graded and counting toward your course or module mark.

By contrast:

Everyday homework for school pupils is more often Hausaufgabe(n).

So in many academic contexts, Hausarbeit is best translated as term paper, seminar paper, or academic paper for a course, not simple homework.

Why is beschreiben in the present tense when we’re talking about a paper’s structure, which seems future or static?

German frequently uses the present tense:

For future actions when the time is clear from context:
- Morgen beschreiben wir unsere Methode. – Tomorrow we will describe…
For “timeless” descriptions of texts, works, and structures:
- In Kapitel 3 analysiert der Autor die Daten.
- Im Fazit fassen wir die wichtigsten Ergebnisse zusammen.

So:

Im Hauptteil der Hausarbeit beschreiben wir unsere Methode genau.

is a standard “text present” (like English uses in literary analysis: “In chapter 3 the author describes …”).

You could also say werden … beschreiben, but in outlines and descriptions of a paper’s structure, the present tense is much more idiomatic:

Im Folgenden erläutern wir …
Zunächst stellen wir die Theorie vor, dann analysieren wir die Daten.

Can I use man instead of wir, or a passive form? What is most natural in academic writing?

Possible options:

wir (first-person plural):
- Im Hauptteil der Hausarbeit beschreiben wir unsere Methode genau.
- Very common in modern academic writing, especially when multiple authors or a research group is involved.
- Sounds direct and clear.
man (impersonal “one”):
- Im Hauptteil der Hausarbeit beschreibt man die Methode genau.
- Grammatically fine, but in academic texts it can sound vague or old-fashioned in some disciplines.
passive:
- Im Hauptteil der Hausarbeit wird die Methode genau beschrieben.
- Very common in formal scientific writing, especially when the focus is on the procedure, not the authors.

What is most natural depends on your field and house style:

Many modern style guides encourage wir for clarity and responsibility.
Very formal or impersonal writing often prefers passive.
man is usually the least preferred in careful academic prose.

What are the genders of Hauptteil and Hausarbeit, and how can I tell?

In this sentence:

Hauptteil is masculine: der Hauptteil
- dative singular: dem Hauptteil → im Hauptteil
Hausarbeit is feminine: die Hausarbeit
- genitive singular: der Hausarbeit

Unfortunately, for these particular words, the gender is largely lexical: you simply have to learn them with their articles:

der Hauptteil
die Hausarbeit

Some hints (not strict rules):

Many -arbeit words are feminine: die Arbeit, die Hausarbeit, die Gruppenarbeit.
Composition Haupt + Teil: the head noun Teil is masculine (der Teil), so the compound Hauptteil is also masculine.

In general, always learn nouns as article + word (e.g. die Hausarbeit, not just Hausarbeit).

Why is there no article before Methode beyond unsere? Could I say die Methode instead?

In unsere Methode, the possessive determiner unsere already plays the role of the article:

In German, you normally do not use a separate definite or indefinite article together with a possessive determiner.
So you say:
- unsere Methode, meine Idee, sein Auto
- not die unsere Methode, eine meine Idee, etc.

Could you say die Methode instead?

unsere Methode = our method (specifically belonging to us/this paper).
die Methode = the method (some definite method, but without the explicit idea of ownership).

In an academic paper where you are presenting your own approach, unsere Methode is more natural, because it emphasizes that this is the method you (the authors) use or propose. If the context already makes clear that “the method” is the one introduced by you, die Methode might also work, but it sounds slightly less explicit.